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Computer Bridge: Single-suit Plans in Notrump II
(Page of 9)

I. Introduction

My article "Single-suit Plans in Notrump" (https://bridgewinners.com/article/view/computer-bridge-single-suit-plans-in-notrump/) ended in a cliffhanger where our computer-bridgeplayer (often called a "bridge robot," and which I have been calling, "Robot") was declaring at notrump, had just seen Dummy tabled, and had used the opening-lead catalog (see https://bridgewinners.com/article/view/computer-bridge-opening-lead-vs-notrump/) and the card-combination catalog (see https://bridgewinners.com/article/view/computer-bridge-card-combinations-in-notrump/) to identify single-suit plans that might help Declarer make the contract and those that might help the defenders set it.

This article continues that deal by examining how single-suit plans might be expanded into multi-trick scenarios to identify the circumstances in which these various plans succeed.

II. Terminology

I have adapted from American football the following terms and assigned them bridge meanings:

  • Possession—one or more consecutive tricks in which the same side leads
  • Series—one or more consecutive tricks in which the same side leads a given suit

III. Situation Facing Declarer (Robot)

The earlier Single-suit Plans article left Robot as Declarer in the following setting:

West
Dummy
AQ10
AQJ
1098
J1042
East
Robot
KJ
K5
A7652
Q653
W
N
E
S
1NT
P
3NT
P
P
P
D
3NT South
NS: 0 EW: 0
8
1

From the declaring side's holdings, the opening lead, the defenders' agreement, and the opening-lead database, Robot had inferred that West led from a worthless tripleton and could hold the following in the other suits:

  • Spades (x < 10): xx, xxx
  • Diamonds (x in {3, 4}): KQx, KJx, Kxx,* QJx, Qxx,* Jxx,* KQxx, KJxx
  • Clubs (x in {7, 8, 9}): AKx, Axx, Kxx, xxx, AKxx, Axxx, Kxxx

Robot also inferred that West's distribution would be 3=3=4=4 minus one (except for hearts), that is, 3=3=4=3, 3=3=3=4, or 2=3=4=4.

Robot then considered for each suit, the single-suit plans available (if any) and how each would fare against the various inferred layouts for that suit. For example, Robot inferred that attacking diamonds by leading to the diamond ace on the first round and continuing diamonds should ultimately generate two winners for the declaring side, at a cost of generating two defensive winners. Robot also inferred that attacking clubs by leading the first round toward the queen and subsequently leading toward Dummy should ultimately generate two winners for the declaring side and no defensive winners (being that the defenders' club ace and king were already masters).

Robot also used the card-combination catalog to determine the defenders' potential for generating winners in each of the four suits.

* Omitted (in error, I believe) from this article's predecessor.

IV. The Declaring Side Attacks Diamonds Only

West
Dummy
AQ10
AQJ
1098
J1042
East
Robot
KJ
K5
A7652
Q653
W
N
E
S
1NT
P
3NT
P
P
P
D
3NT South
NS: 0 EW: 0
8
1

When assessing single-suit plans, it is assumed that whenever the declaring side crosses to the other hand, it is to an original (dealt) master. Lines in which a cross would be effected by finesse in another suit will be considered multi-suit plans.

Consider a single-suit plan that starts with leading a diamond from Dummy for a finesse. Robot would consider how this plan would fare in each of the diamond layouts consistent with Robot's inferences on the previous page. One such layout is where LHO holds KJxx. Against that layout, this plan could involve four series in which diamonds are led:

  1. Leading the suit from Dummy and capturing RHO's queen with the ace, and then leading the second round to one of LHO's honors.
  2. Leading the third round from either hand (ducked by LHO and won in Dummy).
  3. Leading the fourth round from the closed hand to force out LHO's remaining honor.
  4. Leading the fifth-round winner from the closed hand.

 

This scenario would require that the declaring side's non-diamond tricks that are just prior to its diamond series be taken in the following hands:

  1. Just before the first round of diamonds—won by Dummy (in order to take a diamond finesse).
  2. Just before the third round of diamonds—won by either hand (being that Dummy's remaining diamond would be one of several of the declaring side's highest equals in the suit).
  3. Between the third and fourth rounds of diamonds—won by the closed hand (being that Dummy would have no fourth diamond to lead).
  4. Just before the fifth round of diamonds—won by the closed hand.

 

At the moment that Dummy is tabled, the single-dummy layout can provide the two outside entries to the closed hand that this single-suit plan would require: the spade king and the heart king. (This information would be included in what Robot would obtain from the card-combination catalog regarding the defenders' prospects in attacking a given suit.) Robot would need to then analyze whether the diamond winners could be established before the defenders' setting trick and whether there would be transportation to cash them.

V. Analysis of a Single-suit Plan for Attacking Diamonds

West
Dummy
AQ10
AQJ
1098
J1042
East
Robot
KJ
K5
A7652
Q653
W
N
E
S
1NT
P
3NT
P
P
P
D
3NT South
NS: 0 EW: 0
8
1

Robot would assume that the defenders will continue attacking hearts—the suit that the card-combination catalog indicates would be the defenders' most fruitful one outside of diamonds. The four series that constitute the branch of the single-suit plan where LHO holds KJxx would now be interleaved with the defenders' anticipated heart leads. At each step, Robot would determine whether the declaring side had run out of outside entries, the defending side would have cashed its winners for a set, or the declaring side would have cashed its winners for a make. The resulting possessions are listed below:

Possession E/W-1. LHO opening-leads the heart eight, won by Dummy's jack. (In a suit [such as hearts on this deal] where the declaring side holds more master-equals than it does cards in its longer holding, Robot wins the opponents' lead in the desired hand if possible, to preserve entries. And Robot tries to allow for either hand to win a subsequent trick in the suit, and hence would refrain from winning with the ace at Trick 1.)

Trick E/W-1a. LHO leads the heart eight, won by Dummy's jack. (The following would be counted at the conclusion of this trick.)

  • Tricks won by the declaring side: 1
  • Masters held by the declaring side: 6 (For example, the master count in spades is 3 rather than 5, to reflect the number of tricks that masters can take.)
  • Tricks won by the defending side: 0
  • Masters held by the defending side: 2
  • Remaining outside entries to the closed hand: 2

Possession N/S-1. Lead a diamond from Dummy and capture RHO's queen with the ace, and then lead a second round, won by one of LHO's honors.

Trick N/S-1a. Dummy leads a diamond to RHO's queen and Declarer's ace.

  • Tricks won by the declaring side: 2
  • Masters held by the declaring side: 5
  • Tricks won by the defending side: 0
  • Masters held by the defending side: 4 (The defenders' king and jack of diamonds become masters due to the expenditure of Declarer's ace.)
  • Remaining outside entries to the closed hand: 2

Trick N/S-1b. Declarer leads a diamond, taken by one of LHO's honors.

  • Tricks won by the declaring side: 2
  • Masters held by the declaring side: 5
  • Tricks won by the defending side: 1
  • Masters held by the defending side: 3
  • Remaining outside entries to the closed hand: 2

Possession E/W-2. LHO leads another heart, won by Dummy's ace (although it's been noted that winning this trick in the closed hand would be equivalent).

Trick E/W-2a. LHO leads another heart, won by Dummy's ace.

  • Tricks won by the declaring side: 3
  • Masters held by the declaring side: 4
  • Tricks won by the defending side: 1
  • Masters held by the defending side: 3
  • Remaining outside entries to the closed hand: 1 (being that the heart king just fell)

Possession N/S-2. Lead the third round of diamonds from Dummy (ducked by LHO), cross to the closed hand, and lead the fourth round of diamonds.

Trick N/S-2a. Dummy leads a diamond, ducked by LHO. (There would be another scenario for where LHO rises on this trick, but in evaluating how the present single-suit plan fares against KJxx with LHO, Robot would count only the scenario where LHO ducks, because (1) it makes the declaring side expend more entries in order to try to cash the contract-fulfilling trick, and (2) the scenarios indicate that rising would not allow the defending side to win the race to determine make or set.)

  • Tricks won by the declaring side: 4
  • Masters held by the declaring side: 4
  • Tricks won by the defending side: 1
  • Masters held by the defending side: 3
  • Remaining outside entries to the closed hand: 1

Trick N/S-2b. Dummy leads a spade to Declarer's king.

  • Tricks won by the declaring side: 5
  • Masters held by the declaring side: 3
  • Tricks won by the defending side: 1
  • Masters held by the defending side: 3
  • Remaining outside entries to the closed hand: 0

Trick N/S-2c. Declarer leads a diamond to LHO's remaining honor.

  • Tricks won by the declaring side: 5
  • Masters held by the declaring side: 4 (being that the fifth diamond is now established)
  • Tricks won by the defending side: 2
  • Masters held by the defending side: 2
  • Remaining outside entries to the closed hand: 0

Possession E/W-3. LHO leads yet another heart, won by Dummy's queen.

Trick E/W-3a. LHO leads a heart, won by Dummy's queen.

  • Tricks won by the declaring side: 6
  • Masters held by the declaring side: 3
  • Tricks won by the defending side: 2
  • Masters held by the defending side: 4 (being that RHO's inferred long hearts are now established)
  • Remaining outside entries to the closed hand: 0

Possession N/S-3. Cross to the closed hand and lead the (winning) fifth round of diamonds from the closed hand.

Trick N/S-3a. Cross to the closed hand—oops, no outside entry for that.

Robot therefore concludes that diamonds alone cannot fulfill the contract against KJxx with LHO.

VI. Robot's Conclusion about Single-suit Plans for Attacking Diamonds

West
Dummy
AQ10
AQJ
1098
J1042
East
Robot
KJ
K5
A7652
Q653
W
N
E
S
1NT
P
3NT
P
P
P
D
3NT South
NS: 0 EW: 0
8
1

After examining how the various single-suit plans for attacking diamonds would fare against the inferred layouts of the diamond suit, Robot would conclude that none of these plans succeeds when West holds KQxx or KJxx and ducks the third round of the suit. Before examining multi-suit plans, Robot would consider whether a different suit (clubs, in this case) might suffice.

VII. The Declaring Side Attacks Clubs Only

West
Dummy
AQ10
AQJ
1098
J1042
East
Robot
KJ
K5
A7652
Q653
W
N
E
S
1NT
P
3NT
P
P
P
D
3NT South
NS: 0 EW: 0
8
1

Robot now considers a single-suit plan that starts with leading a club from Dummy toward Declarer's queen. If LHO holds AKxx (x in {7, 8, 9}), this plan could involve four series in which clubs are led:

  1. Leading the suit from Dummy to Declarer's queen and LHO's ace or king.
  2. Leading the second round from the closed hand (ducked by LHO and won by Dummy's ten).
  3. Leading the third round from the closed hand (won by LHO's remaining honor).
  4. Leading (or leading to) the fourth-round-winning jack in Dummy.

 

This scenario would require that the declaring side's non-club tricks that are just prior to its club series be taken in the following hands:

  1. Just before the first round of clubs—won by Dummy.
  2. Just before the second round of clubs—won by the closed hand.
  3. Just before the third round of clubs—won by the closed hand.
  4. Just before the fourth round of clubs—won by either hand.

 

At the moment that Dummy is tabled, the single-dummy layout can provide the two outside entries to the closed hand that this single-suit plan would require: the spade king and the heart king. Robot would need to then analyze whether the club winners could be established before the defenders' setting trick and whether there would be transportation to cash the former.

VIII. Analysis of a Single-suit Plan for Attacking Clubs

West
Dummy
AQ10
AQJ
1098
J1042
East
Robot
KJ
K5
A7652
Q653
W
N
E
S
1NT
P
3NT
P
P
P
D
3NT South
NS: 0 EW: 0
8
1

Hearts and diamonds are the two suits that the card-combination catalog indicates would be the defenders' most fruitful ones outside of clubs. However, Robot's inferences indicate that neither suit alone could provide the defenders with sufficient winners to set the contract. Therefore, Robot will assume that the defenders continue attacking hearts, the suit that was opening-led. The four series that constitute the branch of the single-suit plan discussed on the previous page, where LHO holds AKxx (x in {7, 8, 9}), would now be interleaved with the defenders' anticipated heart leads. The resulting possessions are listed below:

Possession E/W-1. LHO leads the heart eight, won by Dummy's jack.

Trick E/W-1a. LHO leads the heart eight, won by Dummy's jack.

  • Tricks won by the declaring side: 1
  • Masters held by the declaring side: 6
  • Tricks won by the defending side: 0
  • Masters held by the defending side: 2
  • Remaining outside entries to the closed hand: 3 (including the diamond ace)

Possession N/S-1. Lead a low club from Dummy to Declarer's queen and LHO's ace or king.

Trick N/S-1a. Dummy leads a low club to Declarer's queen and LHO's ace or king.

  • Tricks won by the declaring side: 1
  • Masters held by the declaring side: 6
  • Tricks won by the defending side: 1
  • Masters held by the defending side: 1
  • Remaining outside entries to the closed hand: 3

Possession E/W-2. LHO leads another heart, Dummy's queen overtaken by Declarer's king.

Trick E/W-2a. LHO leads a heart, won by Declarer's king.

  • Tricks won by the declaring side: 2
  • Masters held by the declaring side: 5
  • Tricks won by the defending side: 1
  • Masters held by the defending side: 1
  • Remaining outside entries to the closed hand: 2

Possession N/S-2. Declarer leads a low club toward the ten (which holds), the closed hand is reentered, and a low club is led toward the jack (taken by LHO's remaining honor).

Trick N/S-2a. Declarer leads a low club toward the ten (which holds).

  • Tricks won by the declaring side: 3
  • Masters held by the declaring side: 5
  • Tricks won by the defending side: 1
  • Masters held by the defending side: 1
  • Remaining outside entries to the closed hand: 2

Trick N/S-2b. Dummy leads a spade to Declarer's king. (Robot prefers spades to diamonds here because crossing in the former leaves more masters in the suit: two versus zero.)

  • Tricks won by the declaring side: 4
  • Masters held by the declaring side: 4
  • Tricks won by the defending side: 1
  • Masters held by the defending side: 1
  • Remaining outside entries to the closed hand: 1

Trick N/S-2c. Declarer leads a club toward the jack (taken by LHO's remaining honor).

  • Tricks won by the declaring side: 4
  • Masters held by the declaring side: 5 (counting Dummy's club jack)
  • Tricks won by the defending side: 2
  • Masters held by the defending side: 0
  • Remaining outside entries to the closed hand: 1

Possession E/W-3. LHO leads yet another heart, won by Dummy's ace.

Trick E/W-3a. LHO leads a heart, won by Dummy's ace.

  • Tricks won by the declaring side: 5
  • Masters held by the declaring side: 4
  • Tricks won by the defending side: 2
  • Masters held by the defending side: 2 (the two long hearts inferred as remaining with RHO)
  • Remaining outside entries to the closed hand: 1

Possession N/S-3. Cash out.

Tricks N/S-3a through N/S-3d. Robot determines that it has the entries to cash out.

Robot therefore concludes that this single-suit plan for attacking clubs is sufficient to fulfill the contract against AKxx (x in {7, 8, 9}) with LHO.

IX. Robot's Conclusions about Single-suit Plans

West
Dummy
AQ10
AQJ
1098
J1042
East
Robot
KJ
K5
A7652
Q653
W
N
E
S
1NT
P
3NT
P
P
P
D
3NT South
NS: 0 EW: 0
8
1

After examining how the various single-suit plans for attacking clubs would fare against the inferred layouts of the club suit, Robot would conclude that the plan examined on the previous two pages would succeed on any club layout consistent with Robot's earlier inferences.

Therefore, Robot would forgo examining any multi-suit plans and would be ready to call for the heart jack from Dummy at Trick 1.

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